/*
 * This software is a cooperative product of The MathWorks and the National
 * Institute of Standards and Technology (NIST) which has been released to the
 * public domain. Neither The MathWorks nor NIST assumes any responsibility
 * whatsoever for its use by other parties, and makes no guarantees, expressed
 * or implied, about its quality, reliability, or any other characteristic.
 */

/*
 * Matrix.java
 * Copyright (C) 1999 The Mathworks and NIST and 2005 University of Waikato,
 *               Hamilton, New Zealand
 *
 */

package weka.core.matrix;

import java.io.BufferedReader;
import java.io.LineNumberReader;
import java.io.PrintWriter;
import java.io.Reader;
import java.io.Serializable;
import java.io.StreamTokenizer;
import java.io.StringReader;
import java.io.StringWriter;
import java.io.Writer;
import java.text.DecimalFormat;
import java.text.DecimalFormatSymbols;
import java.text.NumberFormat;
import java.util.Locale;
import java.util.StringTokenizer;

import weka.core.Utils;

/**
 * Jama = Java Matrix class.
 * <P>
 * The Java Matrix Class provides the fundamental operations of numerical linear
 * algebra. Various constructors create Matrices from two dimensional arrays of
 * double precision floating point numbers. Various "gets" and "sets" provide
 * access to submatrices and matrix elements. Several methods implement basic
 * matrix arithmetic, including matrix addition and multiplication, matrix
 * norms, and element-by-element array operations. Methods for reading and
 * printing matrices are also included. All the operations in this version of
 * the Matrix Class involve real matrices. Complex matrices may be handled in a
 * future version.
 * <P>
 * Five fundamental matrix decompositions, which consist of pairs or triples of
 * matrices, permutation vectors, and the like, produce results in five
 * decomposition classes. These decompositions are accessed by the Matrix class
 * to compute solutions of simultaneous linear equations, determinants, inverses
 * and other matrix functions. The five decompositions are:
 * <P>
 * <UL>
 * <LI>Cholesky Decomposition of symmetric, positive definite matrices.
 * <LI>LU Decomposition of rectangular matrices.
 * <LI>QR Decomposition of rectangular matrices.
 * <LI>Singular Value Decomposition of rectangular matrices.
 * <LI>Eigenvalue Decomposition of both symmetric and nonsymmetric square
 * matrices.
 * </UL>
 * <DL>
 * <DT><B>Example of use:</B></DT>
 * <P>
 * <DD>Solve a linear system A x = b and compute the residual norm, ||b - A x||.
 * <P>
 * 
 * <PRE>
 * double[][] vals = { { 1., 2., 3 }, { 4., 5., 6. }, { 7., 8., 10. } };
 * Matrix A = new Matrix(vals);
 * Matrix b = Matrix.random(3, 1);
 * Matrix x = A.solve(b);
 * Matrix r = A.times(x).minus(b);
 * double rnorm = r.normInf();
 * </PRE>
 * 
 * </DD>
 * </DL>
 * <p/>
 * Adapted from the
 * <a href="http://math.nist.gov/javanumerics/jama/" target="_blank">JAMA</a>
 * package. Additional methods are tagged with the <code>@author</code> tag.
 * 
 * @author The Mathworks and NIST
 * @author Fracpete (fracpete at waikato dot ac dot nz)
 * @version $Revision$
 */
public class Matrix implements Cloneable, Serializable {

    /** for serialization */
    private static final long serialVersionUID = 7856794138418366180L;

    /**
     * Array for internal storage of elements.
     * 
     * @serial internal array storage.
     */
    protected double[][] A;

    /**
     * Row and column dimensions.
     * 
     * @serial row dimension.
     * @serial column dimension.
     */
    protected int m, n;

    /**
     * Construct an m-by-n matrix of zeros.
     * 
     * @param m Number of rows.
     * @param n Number of colums.
     */
    public Matrix(int m, int n) {
        this.m = m;
        this.n = n;
        A = new double[m][n];
    }

    /**
     * Construct an m-by-n constant matrix.
     * 
     * @param m Number of rows.
     * @param n Number of colums.
     * @param s Fill the matrix with this scalar value.
     */
    public Matrix(int m, int n, double s) {
        this.m = m;
        this.n = n;
        A = new double[m][n];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = s;
            }
        }
    }

    /**
     * Construct a matrix from a 2-D array.
     * 
     * @param A Two-dimensional array of doubles.
     * @throws IllegalArgumentException All rows must have the same length
     * @see #constructWithCopy
     */
    public Matrix(double[][] A) {
        m = A.length;
        n = A[0].length;
        for (int i = 0; i < m; i++) {
            if (A[i].length != n) {
                throw new IllegalArgumentException("All rows must have the same length.");
            }
        }
        this.A = A;
    }

    /**
     * Construct a matrix quickly without checking arguments.
     * 
     * @param A Two-dimensional array of doubles.
     * @param m Number of rows.
     * @param n Number of colums.
     */
    public Matrix(double[][] A, int m, int n) {
        this.A = A;
        this.m = m;
        this.n = n;
    }

    /**
     * Construct a matrix from a one-dimensional packed array
     * 
     * @param vals One-dimensional array of doubles, packed by columns (ala
     *             Fortran).
     * @param m    Number of rows.
     * @throws IllegalArgumentException Array length must be a multiple of m.
     */
    public Matrix(double vals[], int m) {
        this.m = m;
        n = (m != 0 ? vals.length / m : 0);
        if (m * n != vals.length) {
            throw new IllegalArgumentException("Array length must be a multiple of m.");
        }
        A = new double[m][n];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = vals[i + j * m];
            }
        }
    }

    /**
     * Reads a matrix from a reader. The first line in the file should contain the
     * number of rows and columns. Subsequent lines contain elements of the matrix.
     * (FracPete: taken from old weka.core.Matrix class)
     * 
     * @param r the reader containing the matrix
     * @throws Exception if an error occurs
     * @see #write(Writer)
     */
    public Matrix(Reader r) throws Exception {
        LineNumberReader lnr = new LineNumberReader(r);
        String line;
        int currentRow = -1;

        while ((line = lnr.readLine()) != null) {

            // Comments
            if (line.startsWith("%")) {
                continue;
            }

            StringTokenizer st = new StringTokenizer(line);
            // Ignore blank lines
            if (!st.hasMoreTokens()) {
                continue;
            }

            if (currentRow < 0) {
                int rows = Integer.parseInt(st.nextToken());
                if (!st.hasMoreTokens()) {
                    throw new Exception("Line " + lnr.getLineNumber() + ": expected number of columns");
                }

                int cols = Integer.parseInt(st.nextToken());
                A = new double[rows][cols];
                m = rows;
                n = cols;
                currentRow++;
                continue;

            } else {
                if (currentRow == getRowDimension()) {
                    throw new Exception("Line " + lnr.getLineNumber() + ": too many rows provided");
                }

                for (int i = 0; i < getColumnDimension(); i++) {
                    if (!st.hasMoreTokens()) {
                        throw new Exception("Line " + lnr.getLineNumber() + ": too few matrix elements provided");
                    }

                    set(currentRow, i, Double.valueOf(st.nextToken()).doubleValue());
                }
                currentRow++;
            }
        }

        if (currentRow == -1) {
            throw new Exception("Line " + lnr.getLineNumber() + ": expected number of rows");
        } else if (currentRow != getRowDimension()) {
            throw new Exception("Line " + lnr.getLineNumber() + ": too few rows provided");
        }
    }

    /**
     * Construct a matrix from a copy of a 2-D array.
     * 
     * @param A Two-dimensional array of doubles.
     * @throws IllegalArgumentException All rows must have the same length
     */
    public static Matrix constructWithCopy(double[][] A) {
        int m = A.length;
        int n = A[0].length;
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            if (A[i].length != n) {
                throw new IllegalArgumentException("All rows must have the same length.");
            }
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j];
            }
        }
        return X;
    }

    /**
     * Make a deep copy of a matrix
     */
    public Matrix copy() {
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j];
            }
        }
        return X;
    }

    /**
     * Clone the Matrix object.
     */
    @Override
    public Object clone() {
        return this.copy();
    }

    /**
     * Access the internal two-dimensional array.
     * 
     * @return Pointer to the two-dimensional array of matrix elements.
     */
    public double[][] getArray() {
        return A;
    }

    /**
     * Copy the internal two-dimensional array.
     * 
     * @return Two-dimensional array copy of matrix elements.
     */
    public double[][] getArrayCopy() {
        double[][] C = new double[m][n];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j];
            }
        }
        return C;
    }

    /**
     * Make a one-dimensional column packed copy of the internal array.
     * 
     * @return Matrix elements packed in a one-dimensional array by columns.
     */
    public double[] getColumnPackedCopy() {
        double[] vals = new double[m * n];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                vals[i + j * m] = A[i][j];
            }
        }
        return vals;
    }

    /**
     * Make a one-dimensional row packed copy of the internal array.
     * 
     * @return Matrix elements packed in a one-dimensional array by rows.
     */
    public double[] getRowPackedCopy() {
        double[] vals = new double[m * n];
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                vals[i * n + j] = A[i][j];
            }
        }
        return vals;
    }

    /**
     * Get row dimension.
     * 
     * @return m, the number of rows.
     */
    public int getRowDimension() {
        return m;
    }

    /**
     * Get column dimension.
     * 
     * @return n, the number of columns.
     */
    public int getColumnDimension() {
        return n;
    }

    /**
     * Get a single element.
     * 
     * @param i Row index.
     * @param j Column index.
     * @return A(i,j)
     * @throws ArrayIndexOutOfBoundsException
     */
    public double get(int i, int j) {
        return A[i][j];
    }

    /**
     * Get a submatrix.
     * 
     * @param i0 Initial row index
     * @param i1 Final row index
     * @param j0 Initial column index
     * @param j1 Final column index
     * @return A(i0:i1,j0:j1)
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public Matrix getMatrix(int i0, int i1, int j0, int j1) {
        Matrix X = new Matrix(i1 - i0 + 1, j1 - j0 + 1);
        double[][] B = X.getArray();
        try {
            for (int i = i0; i <= i1; i++) {
                for (int j = j0; j <= j1; j++) {
                    B[i - i0][j - j0] = A[i][j];
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
        return X;
    }

    /**
     * Get a submatrix.
     * 
     * @param r Array of row indices.
     * @param c Array of column indices.
     * @return A(r(:),c(:))
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public Matrix getMatrix(int[] r, int[] c) {
        Matrix X = new Matrix(r.length, c.length);
        double[][] B = X.getArray();
        try {
            for (int i = 0; i < r.length; i++) {
                for (int j = 0; j < c.length; j++) {
                    B[i][j] = A[r[i]][c[j]];
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
        return X;
    }

    /**
     * Get a submatrix.
     * 
     * @param i0 Initial row index
     * @param i1 Final row index
     * @param c  Array of column indices.
     * @return A(i0:i1,c(:))
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public Matrix getMatrix(int i0, int i1, int[] c) {
        Matrix X = new Matrix(i1 - i0 + 1, c.length);
        double[][] B = X.getArray();
        try {
            for (int i = i0; i <= i1; i++) {
                for (int j = 0; j < c.length; j++) {
                    B[i - i0][j] = A[i][c[j]];
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
        return X;
    }

    /**
     * Get a submatrix.
     * 
     * @param r  Array of row indices.
     * @param j0 Initial column index
     * @param j1 Final column index
     * @return A(r(:),j0:j1)
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public Matrix getMatrix(int[] r, int j0, int j1) {
        Matrix X = new Matrix(r.length, j1 - j0 + 1);
        double[][] B = X.getArray();
        try {
            for (int i = 0; i < r.length; i++) {
                for (int j = j0; j <= j1; j++) {
                    B[i][j - j0] = A[r[i]][j];
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
        return X;
    }

    /**
     * Set a single element.
     * 
     * @param i Row index.
     * @param j Column index.
     * @param s A(i,j).
     * @throws ArrayIndexOutOfBoundsException
     */
    public void set(int i, int j, double s) {
        A[i][j] = s;
    }

    /**
     * Set a submatrix.
     * 
     * @param i0 Initial row index
     * @param i1 Final row index
     * @param j0 Initial column index
     * @param j1 Final column index
     * @param X  A(i0:i1,j0:j1)
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public void setMatrix(int i0, int i1, int j0, int j1, Matrix X) {
        try {
            for (int i = i0; i <= i1; i++) {
                for (int j = j0; j <= j1; j++) {
                    A[i][j] = X.get(i - i0, j - j0);
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
    }

    /**
     * Set a submatrix.
     * 
     * @param r Array of row indices.
     * @param c Array of column indices.
     * @param X A(r(:),c(:))
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public void setMatrix(int[] r, int[] c, Matrix X) {
        try {
            for (int i = 0; i < r.length; i++) {
                for (int j = 0; j < c.length; j++) {
                    A[r[i]][c[j]] = X.get(i, j);
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
    }

    /**
     * Set a submatrix.
     * 
     * @param r  Array of row indices.
     * @param j0 Initial column index
     * @param j1 Final column index
     * @param X  A(r(:),j0:j1)
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public void setMatrix(int[] r, int j0, int j1, Matrix X) {
        try {
            for (int i = 0; i < r.length; i++) {
                for (int j = j0; j <= j1; j++) {
                    A[r[i]][j] = X.get(i, j - j0);
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
    }

    /**
     * Set a submatrix.
     * 
     * @param i0 Initial row index
     * @param i1 Final row index
     * @param c  Array of column indices.
     * @param X  A(i0:i1,c(:))
     * @throws ArrayIndexOutOfBoundsException Submatrix indices
     */
    public void setMatrix(int i0, int i1, int[] c, Matrix X) {
        try {
            for (int i = i0; i <= i1; i++) {
                for (int j = 0; j < c.length; j++) {
                    A[i][c[j]] = X.get(i - i0, j);
                }
            }
        } catch (ArrayIndexOutOfBoundsException e) {
            throw new ArrayIndexOutOfBoundsException("Submatrix indices");
        }
    }

    /**
     * Returns true if the matrix is symmetric. (FracPete: taken from old
     * weka.core.Matrix class)
     * 
     * @return boolean true if matrix is symmetric.
     */
    public boolean isSymmetric() {
        int nr = A.length, nc = A[0].length;
        if (nr != nc) {
            return false;
        }

        for (int i = 0; i < nc; i++) {
            for (int j = 0; j < i; j++) {
                if (A[i][j] != A[j][i]) {
                    return false;
                }
            }
        }
        return true;
    }

    /**
     * returns whether the matrix is a square matrix or not.
     * 
     * @return true if the matrix is a square matrix
     */
    public boolean isSquare() {
        return (getRowDimension() == getColumnDimension());
    }

    /**
     * Matrix transpose.
     * 
     * @return A'
     */
    public Matrix transpose() {
        Matrix X = new Matrix(n, m);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[j][i] = A[i][j];
            }
        }
        return X;
    }

    /**
     * One norm
     * 
     * @return maximum column sum.
     */
    public double norm1() {
        double f = 0;
        for (int j = 0; j < n; j++) {
            double s = 0;
            for (int i = 0; i < m; i++) {
                s += Math.abs(A[i][j]);
            }
            f = Math.max(f, s);
        }
        return f;
    }

    /**
     * Two norm
     * 
     * @return maximum singular value.
     */
    public double norm2() {
        return (new SingularValueDecomposition(this).norm2());
    }

    /**
     * Infinity norm
     * 
     * @return maximum row sum.
     */
    public double normInf() {
        double f = 0;
        for (int i = 0; i < m; i++) {
            double s = 0;
            for (int j = 0; j < n; j++) {
                s += Math.abs(A[i][j]);
            }
            f = Math.max(f, s);
        }
        return f;
    }

    /**
     * Frobenius norm
     * 
     * @return sqrt of sum of squares of all elements.
     */
    public double normF() {
        double f = 0;
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                f = Maths.hypot(f, A[i][j]);
            }
        }
        return f;
    }

    /**
     * Unary minus
     * 
     * @return -A
     */
    public Matrix uminus() {
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = -A[i][j];
            }
        }
        return X;
    }

    /**
     * C = A + B
     * 
     * @param B another matrix
     * @return A + B
     */
    public Matrix plus(Matrix B) {
        checkMatrixDimensions(B);
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j] + B.A[i][j];
            }
        }
        return X;
    }

    /**
     * A = A + B
     * 
     * @param B another matrix
     * @return A + B
     */
    public Matrix plusEquals(Matrix B) {
        checkMatrixDimensions(B);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = A[i][j] + B.A[i][j];
            }
        }
        return this;
    }

    /**
     * C = A - B
     * 
     * @param B another matrix
     * @return A - B
     */
    public Matrix minus(Matrix B) {
        checkMatrixDimensions(B);
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j] - B.A[i][j];
            }
        }
        return X;
    }

    /**
     * A = A - B
     * 
     * @param B another matrix
     * @return A - B
     */
    public Matrix minusEquals(Matrix B) {
        checkMatrixDimensions(B);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = A[i][j] - B.A[i][j];
            }
        }
        return this;
    }

    /**
     * Element-by-element multiplication, C = A.*B
     * 
     * @param B another matrix
     * @return A.*B
     */
    public Matrix arrayTimes(Matrix B) {
        checkMatrixDimensions(B);
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j] * B.A[i][j];
            }
        }
        return X;
    }

    /**
     * Element-by-element multiplication in place, A = A.*B
     * 
     * @param B another matrix
     * @return A.*B
     */
    public Matrix arrayTimesEquals(Matrix B) {
        checkMatrixDimensions(B);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = A[i][j] * B.A[i][j];
            }
        }
        return this;
    }

    /**
     * Element-by-element right division, C = A./B
     * 
     * @param B another matrix
     * @return A./B
     */
    public Matrix arrayRightDivide(Matrix B) {
        checkMatrixDimensions(B);
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = A[i][j] / B.A[i][j];
            }
        }
        return X;
    }

    /**
     * Element-by-element right division in place, A = A./B
     * 
     * @param B another matrix
     * @return A./B
     */
    public Matrix arrayRightDivideEquals(Matrix B) {
        checkMatrixDimensions(B);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = A[i][j] / B.A[i][j];
            }
        }
        return this;
    }

    /**
     * Element-by-element left division, C = A.\B
     * 
     * @param B another matrix
     * @return A.\B
     */
    public Matrix arrayLeftDivide(Matrix B) {
        checkMatrixDimensions(B);
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = B.A[i][j] / A[i][j];
            }
        }
        return X;
    }

    /**
     * Element-by-element left division in place, A = A.\B
     * 
     * @param B another matrix
     * @return A.\B
     */
    public Matrix arrayLeftDivideEquals(Matrix B) {
        checkMatrixDimensions(B);
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = B.A[i][j] / A[i][j];
            }
        }
        return this;
    }

    /**
     * Multiply a matrix by a scalar, C = s*A
     * 
     * @param s scalar
     * @return s*A
     */
    public Matrix times(double s) {
        Matrix X = new Matrix(m, n);
        double[][] C = X.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                C[i][j] = s * A[i][j];
            }
        }
        return X;
    }

    /**
     * Multiply a matrix by a scalar in place, A = s*A
     * 
     * @param s scalar
     * @return replace A by s*A
     */
    public Matrix timesEquals(double s) {
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                A[i][j] = s * A[i][j];
            }
        }
        return this;
    }

    /**
     * Linear algebraic matrix multiplication, A * B
     * 
     * @param B another matrix
     * @return Matrix product, A * B
     * @throws IllegalArgumentException Matrix inner dimensions must agree.
     */
    public Matrix times(Matrix B) {
        if (B.m != n) {
            throw new IllegalArgumentException("Matrix inner dimensions must agree.");
        }
        Matrix X = new Matrix(m, B.n);
        double[][] C = X.getArray();
        double[] Bcolj = new double[n];
        for (int j = 0; j < B.n; j++) {
            for (int k = 0; k < n; k++) {
                Bcolj[k] = B.A[k][j];
            }
            for (int i = 0; i < m; i++) {
                double[] Arowi = A[i];
                double s = 0;
                for (int k = 0; k < n; k++) {
                    s += Arowi[k] * Bcolj[k];
                }
                C[i][j] = s;
            }
        }
        return X;
    }

    /**
     * LU Decomposition
     * 
     * @return LUDecomposition
     * @see LUDecomposition
     */
    public LUDecomposition lu() {
        return new LUDecomposition(this);
    }

    /**
     * QR Decomposition
     * 
     * @return QRDecomposition
     * @see QRDecomposition
     */
    public QRDecomposition qr() {
        return new QRDecomposition(this);
    }

    /**
     * Cholesky Decomposition
     * 
     * @return CholeskyDecomposition
     * @see CholeskyDecomposition
     */
    public CholeskyDecomposition chol() {
        return new CholeskyDecomposition(this);
    }

    /**
     * Singular Value Decomposition
     * 
     * @return SingularValueDecomposition
     * @see SingularValueDecomposition
     */
    public SingularValueDecomposition svd() {
        return new SingularValueDecomposition(this);
    }

    /**
     * Eigenvalue Decomposition
     * 
     * @return EigenvalueDecomposition
     * @see EigenvalueDecomposition
     */
    public EigenvalueDecomposition eig() {
        return new EigenvalueDecomposition(this);
    }

    /**
     * Solve A*X = B
     * 
     * @param B right hand side
     * @return solution if A is square, least squares solution otherwise
     */
    public Matrix solve(Matrix B) {
        return (m == n ? (new LUDecomposition(this)).solve(B) : (new QRDecomposition(this)).solve(B));
    }

    /**
     * Solve X*A = B, which is also A'*X' = B'
     * 
     * @param B right hand side
     * @return solution if A is square, least squares solution otherwise.
     */
    public Matrix solveTranspose(Matrix B) {
        return transpose().solve(B.transpose());
    }

    /**
     * Matrix inverse or pseudoinverse
     * 
     * @return inverse(A) if A is square, pseudoinverse otherwise.
     */
    public Matrix inverse() {
        return solve(identity(m, m));
    }

    /**
     * returns the square root of the matrix, i.e., X from the equation X*X =
     * A.<br/>
     * Steps in the Calculation (see <a href=
     * "http://www.mathworks.com/access/helpdesk/help/techdoc/ref/sqrtm.html" target
     * ="blank"><code>sqrtm</code></a> in Matlab):<br/>
     * <ol>
     * <li>perform eigenvalue decomposition<br/>
     * [V,D]=eig(A)</li>
     * <li>take the square root of all elements in D (only the ones with positive
     * sign are considered for further computation)<br/>
     * S=sqrt(D)</li>
     * <li>calculate the root<br/>
     * X=V*S/V, which can be also written as X=(V'\(V*S)')'</li>
     * </ol>
     * <p/>
     * <b>Note:</b> since this method uses other high-level methods, it generates
     * several instances of matrices. This can be problematic with large matrices.
     * <p/>
     * Examples:
     * <ol>
     * <li>
     * 
     * <pre>
     *  X =
     *   5   -4    1    0    0
     *  -4    6   -4    1    0
     *   1   -4    6   -4    1
     *   0    1   -4    6   -4
     *   0    0    1   -4    5
     * 
     *  sqrt(X) =
     *   2   -1   -0   -0   -0 
     *  -1    2   -1    0   -0 
     *   0   -1    2   -1    0 
     *  -0    0   -1    2   -1 
     *  -0   -0   -0   -1    2 
     *  
     *  Matrix m = new Matrix(new double[][]{{5,-4,1,0,0},{-4,6,-4,1,0},{1,-4,6,-4,1},{0,1,-4,6,-4},{0,0,1,-4,5}});
     * </pre>
     * 
     * </li>
     * <li>
     * 
     * <pre>
     *  X =
     *   7   10
     *  15   22
     *  
     *  sqrt(X) =
     *  1.5667    1.7408
     *  2.6112    4.1779
     * 
     *  Matrix m = new Matrix(new double[][]{{7, 10},{15, 22}});
     * </pre>
     * 
     * </li>
     * </ol>
     * 
     * @return sqrt(A)
     */
    public Matrix sqrt() {
        EigenvalueDecomposition evd;
        Matrix s;
        Matrix v;
        Matrix d;
        Matrix result;
        Matrix a;
        Matrix b;
        int i;
        int n;

        result = null;

        // eigenvalue decomp.
        // [V, D] = eig(A) with A = this
        evd = this.eig();
        v = evd.getV();
        d = evd.getD();

        // S = sqrt of cells of D
        s = new Matrix(d.getRowDimension(), d.getColumnDimension());
        for (i = 0; i < s.getRowDimension(); i++) {
            for (n = 0; n < s.getColumnDimension(); n++) {
                s.set(i, n, StrictMath.sqrt(d.get(i, n)));
            }
        }

        // to calculate:
        // result = V*S/V
        //
        // with X = B/A
        // and B/A = (A'\B')'
        // and V=A and V*S=B
        // we get
        // result = (V'\(V*S)')'
        //
        // A*X = B
        // X = A\B
        // which is
        // X = A.solve(B)
        //
        // with A=V' and B=(V*S)'
        // we get
        // X = V'.solve((V*S)')
        // or
        // result = X'
        //
        // which is in full length
        // result = (V'.solve((V*S)'))'
        a = v.inverse();
        b = v.times(s).inverse();
        v = null;
        d = null;
        evd = null;
        s = null;
        result = a.solve(b).inverse();

        return result;
    }

    /**
     * Performs a (ridged) linear regression. (FracPete: taken from old
     * weka.core.Matrix class)
     * 
     * @param y     the dependent variable vector
     * @param ridge the ridge parameter
     * @return the coefficients
     * @throws IllegalArgumentException if not successful
     */
    public LinearRegression regression(Matrix y, double ridge) {
        return new LinearRegression(this, y, ridge);
    }

    /**
     * Performs a weighted (ridged) linear regression. (FracPete: taken from old
     * weka.core.Matrix class)
     * 
     * @param y     the dependent variable vector
     * @param w     the array of data point weights
     * @param ridge the ridge parameter
     * @return the coefficients
     * @throws IllegalArgumentException if the wrong number of weights were
     *                                  provided.
     */
    public final LinearRegression regression(Matrix y, double[] w, double ridge) {
        return new LinearRegression(this, y, w, ridge);
    }

    /**
     * Matrix determinant
     * 
     * @return determinant
     */
    public double det() {
        return new LUDecomposition(this).det();
    }

    /**
     * Matrix rank
     * 
     * @return effective numerical rank, obtained from SVD.
     */
    public int rank() {
        return new SingularValueDecomposition(this).rank();
    }

    /**
     * Matrix condition (2 norm)
     * 
     * @return ratio of largest to smallest singular value.
     */
    public double cond() {
        return new SingularValueDecomposition(this).cond();
    }

    /**
     * Matrix trace.
     * 
     * @return sum of the diagonal elements.
     */
    public double trace() {
        double t = 0;
        for (int i = 0; i < Math.min(m, n); i++) {
            t += A[i][i];
        }
        return t;
    }

    /**
     * Generate matrix with random elements
     * 
     * @param m Number of rows.
     * @param n Number of colums.
     * @return An m-by-n matrix with uniformly distributed random elements.
     */
    public static Matrix random(int m, int n) {
        Matrix A = new Matrix(m, n);
        double[][] X = A.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                X[i][j] = Math.random();
            }
        }
        return A;
    }

    /**
     * Generate identity matrix
     * 
     * @param m Number of rows.
     * @param n Number of colums.
     * @return An m-by-n matrix with ones on the diagonal and zeros elsewhere.
     */
    public static Matrix identity(int m, int n) {
        Matrix A = new Matrix(m, n);
        double[][] X = A.getArray();
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                X[i][j] = (i == j ? 1.0 : 0.0);
            }
        }
        return A;
    }

    /**
     * Print the matrix to stdout. Line the elements up in columns with a
     * Fortran-like 'Fw.d' style format.
     * 
     * @param w Column width.
     * @param d Number of digits after the decimal.
     */
    public void print(int w, int d) {
        print(new PrintWriter(System.out, true), w, d);
    }

    /**
     * Print the matrix to the output stream. Line the elements up in columns with a
     * Fortran-like 'Fw.d' style format.
     * 
     * @param output Output stream.
     * @param w      Column width.
     * @param d      Number of digits after the decimal.
     */
    public void print(PrintWriter output, int w, int d) {
        DecimalFormat format = new DecimalFormat();
        format.setDecimalFormatSymbols(new DecimalFormatSymbols(Locale.US));
        format.setMinimumIntegerDigits(1);
        format.setMaximumFractionDigits(d);
        format.setMinimumFractionDigits(d);
        format.setGroupingUsed(false);
        print(output, format, w + 2);
    }

    /**
     * Print the matrix to stdout. Line the elements up in columns. Use the format
     * object, and right justify within columns of width characters. Note that is
     * the matrix is to be read back in, you probably will want to use a
     * NumberFormat that is set to US Locale.
     * 
     * @param format A Formatting object for individual elements.
     * @param width  Field width for each column.
     * @see java.text.DecimalFormat#setDecimalFormatSymbols
     */
    public void print(NumberFormat format, int width) {
        print(new PrintWriter(System.out, true), format, width);
    }

    // DecimalFormat is a little disappointing coming from Fortran or C's printf.
    // Since it doesn't pad on the left, the elements will come out different
    // widths. Consequently, we'll pass the desired column width in as an
    // argument and do the extra padding ourselves.

    /**
     * Print the matrix to the output stream. Line the elements up in columns. Use
     * the format object, and right justify within columns of width characters. Note
     * that is the matrix is to be read back in, you probably will want to use a
     * NumberFormat that is set to US Locale.
     * 
     * @param output the output stream.
     * @param format A formatting object to format the matrix elements
     * @param width  Column width.
     * @see java.text.DecimalFormat#setDecimalFormatSymbols
     */
    public void print(PrintWriter output, NumberFormat format, int width) {
        output.println(); // start on new line.
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                String s = format.format(A[i][j]); // format the number
                int padding = Math.max(1, width - s.length()); // At _least_ 1 space
                for (int k = 0; k < padding; k++) {
                    output.print(' ');
                }
                output.print(s);
            }
            output.println();
        }
        output.println(); // end with blank line.
    }

    /**
     * Read a matrix from a stream. The format is the same the print method, so
     * printed matrices can be read back in (provided they were printed using US
     * Locale). Elements are separated by whitespace, all the elements for each row
     * appear on a single line, the last row is followed by a blank line.
     * <p/>
     * Note: This format differs from the one that can be read via the
     * Matrix(Reader) constructor! For that format, the write(Writer) method is used
     * (from the original weka.core.Matrix class).
     * 
     * @param input the input stream.
     * @see #Matrix(Reader)
     * @see #write(Writer)
     */
    public static Matrix read(BufferedReader input) throws java.io.IOException {
        StreamTokenizer tokenizer = new StreamTokenizer(input);

        // Although StreamTokenizer will parse numbers, it doesn't recognize
        // scientific notation (E or D); however, Double.valueOf does.
        // The strategy here is to disable StreamTokenizer's number parsing.
        // We'll only get whitespace delimited words, EOL's and EOF's.
        // These words should all be numbers, for Double.valueOf to parse.

        tokenizer.resetSyntax();
        tokenizer.wordChars(0, 255);
        tokenizer.whitespaceChars(0, ' ');
        tokenizer.eolIsSignificant(true);
        java.util.Vector<Object> v = new java.util.Vector<Object>();

        // Ignore initial empty lines
        while (tokenizer.nextToken() == StreamTokenizer.TT_EOL) {
            ;
        }
        if (tokenizer.ttype == StreamTokenizer.TT_EOF) {
            throw new java.io.IOException("Unexpected EOF on matrix read.");
        }
        do {
            v.addElement(Double.valueOf(tokenizer.sval)); // Read & store 1st row.
        } while (tokenizer.nextToken() == StreamTokenizer.TT_WORD);

        int n = v.size(); // Now we've got the number of columns!
        double row[] = new double[n];
        for (int j = 0; j < n; j++) {
            row[j] = ((Double) v.elementAt(j)).doubleValue();
        }
        v.removeAllElements();
        v.addElement(row); // Start storing rows instead of columns.
        while (tokenizer.nextToken() == StreamTokenizer.TT_WORD) {
            // While non-empty lines
            v.addElement(row = new double[n]);
            int j = 0;
            do {
                if (j >= n) {
                    throw new java.io.IOException("Row " + v.size() + " is too long.");
                }
                row[j++] = Double.valueOf(tokenizer.sval).doubleValue();
            } while (tokenizer.nextToken() == StreamTokenizer.TT_WORD);
            if (j < n) {
                throw new java.io.IOException("Row " + v.size() + " is too short.");
            }
        }
        int m = v.size(); // Now we've got the number of rows.
        double[][] A = new double[m][];
        v.copyInto(A); // copy the rows out of the vector
        return new Matrix(A);
    }

    /**
     * Check if size(A) == size(B)
     */
    private void checkMatrixDimensions(Matrix B) {
        if (B.m != m || B.n != n) {
            throw new IllegalArgumentException("Matrix dimensions must agree.");
        }
    }

    /**
     * Writes out a matrix. The format can be read via the Matrix(Reader)
     * constructor. (FracPete: taken from old weka.core.Matrix class)
     * 
     * @param w the output Writer
     * @throws Exception if an error occurs
     * @see #Matrix(Reader)
     */
    public void write(Writer w) throws Exception {
        w.write("% Rows\tColumns\n");
        w.write("" + getRowDimension() + "\t" + getColumnDimension() + "\n");
        w.write("% Matrix elements\n");
        for (int i = 0; i < getRowDimension(); i++) {
            for (int j = 0; j < getColumnDimension(); j++) {
                w.write("" + get(i, j) + "\t");
            }
            w.write("\n");
        }
        w.flush();
    }

    /**
     * Converts a matrix to a string. (FracPete: taken from old weka.core.Matrix
     * class)
     * 
     * @return the converted string
     */
    @Override
    public String toString() {
        // Determine the width required for the maximum element,
        // and check for fractional display requirement.
        double maxval = 0;
        boolean fractional = false;
        for (int i = 0; i < getRowDimension(); i++) {
            for (int j = 0; j < getColumnDimension(); j++) {
                double current = get(i, j);
                if (current < 0) {
                    current *= -11;
                }
                if (current > maxval) {
                    maxval = current;
                }
                double fract = Math.abs(current - Math.rint(current));
                if (!fractional && ((Math.log(fract) / Math.log(10)) >= -2)) {
                    fractional = true;
                }
            }
        }
        int width = (int) (Math.log(maxval) / Math.log(10) + (fractional ? 4 : 1));

        StringBuffer text = new StringBuffer();
        for (int i = 0; i < getRowDimension(); i++) {
            for (int j = 0; j < getColumnDimension(); j++) {
                text.append(" ").append(Utils.doubleToString(get(i, j), width, (fractional ? 2 : 0)));
            }
            text.append("\n");
        }

        return text.toString();
    }

    /**
     * converts the Matrix into a single line Matlab string: matrix is enclosed by
     * parentheses, rows are separated by semicolon and single cells by blanks,
     * e.g., [1 2; 3 4].
     * 
     * @return the matrix in Matlab single line format
     */
    public String toMatlab() {
        StringBuffer result;
        int i;
        int n;

        result = new StringBuffer();

        result.append("[");

        for (i = 0; i < getRowDimension(); i++) {
            if (i > 0) {
                result.append("; ");
            }

            for (n = 0; n < getColumnDimension(); n++) {
                if (n > 0) {
                    result.append(" ");
                }
                result.append(Double.toString(get(i, n)));
            }
        }

        result.append("]");

        return result.toString();
    }

    /**
     * creates a matrix from the given Matlab string.
     * 
     * @param matlab the matrix in matlab format
     * @return the matrix represented by the given string
     * @see #toMatlab()
     */
    public static Matrix parseMatlab(String matlab) throws Exception {
        StringTokenizer tokRow;
        StringTokenizer tokCol;
        int rows;
        int cols;
        Matrix result;
        String cells;

        // get content
        cells = matlab.substring(matlab.indexOf("[") + 1, matlab.indexOf("]")).trim();

        // determine dimenions
        tokRow = new StringTokenizer(cells, ";");
        rows = tokRow.countTokens();
        tokCol = new StringTokenizer(tokRow.nextToken(), " ");
        cols = tokCol.countTokens();

        // fill matrix
        result = new Matrix(rows, cols);
        tokRow = new StringTokenizer(cells, ";");
        rows = 0;
        while (tokRow.hasMoreTokens()) {
            tokCol = new StringTokenizer(tokRow.nextToken(), " ");
            cols = 0;
            while (tokCol.hasMoreTokens()) {
                result.set(rows, cols, Double.parseDouble(tokCol.nextToken()));
                cols++;
            }
            rows++;
        }

        return result;
    }

    /**
     * Main method for testing this class.
     */
    public static void main(String[] args) {
        Matrix I;
        Matrix A;
        Matrix B;

        try {
            // Identity
            System.out.println("\nIdentity\n");
            I = Matrix.identity(3, 5);
            System.out.println("I(3,5)\n" + I);

            // basic operations - square
            System.out.println("\nbasic operations - square\n");
            A = Matrix.random(3, 3);
            B = Matrix.random(3, 3);
            System.out.println("A\n" + A);
            System.out.println("B\n" + B);
            System.out.println("A'\n" + A.inverse());
            System.out.println("A^T\n" + A.transpose());
            System.out.println("A+B\n" + A.plus(B));
            System.out.println("A*B\n" + A.times(B));
            System.out.println("X from A*X=B\n" + A.solve(B));

            // basic operations - non square
            System.out.println("\nbasic operations - non square\n");
            A = Matrix.random(2, 3);
            B = Matrix.random(3, 4);
            System.out.println("A\n" + A);
            System.out.println("B\n" + B);
            System.out.println("A*B\n" + A.times(B));

            // sqrt
            System.out.println("\nsqrt (1)\n");
            A = new Matrix(new double[][] { { 5, -4, 1, 0, 0 }, { -4, 6, -4, 1, 0 }, { 1, -4, 6, -4, 1 }, { 0, 1, -4, 6, -4 }, { 0, 0, 1, -4, 5 } });
            System.out.println("A\n" + A);
            System.out.println("sqrt(A)\n" + A.sqrt());

            // sqrt
            System.out.println("\nsqrt (2)\n");
            A = new Matrix(new double[][] { { 7, 10 }, { 15, 22 } });
            System.out.println("A\n" + A);
            System.out.println("sqrt(A)\n" + A.sqrt());
            System.out.println("det(A)\n" + A.det() + "\n");

            // eigenvalue decomp.
            System.out.println("\nEigenvalue Decomposition\n");
            EigenvalueDecomposition evd = A.eig();
            System.out.println("[V,D] = eig(A)");
            System.out.println("- V\n" + evd.getV());
            System.out.println("- D\n" + evd.getD());

            // LU decomp.
            System.out.println("\nLU Decomposition\n");
            LUDecomposition lud = A.lu();
            System.out.println("[L,U,P] = lu(A)");
            System.out.println("- L\n" + lud.getL());
            System.out.println("- U\n" + lud.getU());
            System.out.println("- P\n" + Utils.arrayToString(lud.getPivot()) + "\n");

            // regression
            System.out.println("\nRegression\n");
            B = new Matrix(new double[][] { { 3 }, { 2 } });
            double ridge = 0.5;
            double[] weights = new double[] { 0.3, 0.7 };
            System.out.println("A\n" + A);
            System.out.println("B\n" + B);
            System.out.println("ridge = " + ridge + "\n");
            System.out.println("weights = " + Utils.arrayToString(weights) + "\n");
            System.out.println("A.regression(B, ridge)\n" + A.regression(B, ridge) + "\n");
            System.out.println("A.regression(B, weights, ridge)\n" + A.regression(B, weights, ridge) + "\n");

            // writer/reader
            System.out.println("\nWriter/Reader\n");
            StringWriter writer = new StringWriter();
            A.write(writer);
            System.out.println("A.write(Writer)\n" + writer);
            A = new Matrix(new StringReader(writer.toString()));
            System.out.println("A = new Matrix.read(Reader)\n" + A);

            // Matlab
            System.out.println("\nMatlab-Format\n");
            String matlab = "[ 1   2;3 4 ]";
            System.out.println("Matlab: " + matlab);
            System.out.println("from Matlab:\n" + Matrix.parseMatlab(matlab));
            System.out.println("to Matlab:\n" + Matrix.parseMatlab(matlab).toMatlab());
            matlab = "[1 2 3 4;3 4 5 6;7 8 9 10]";
            System.out.println("Matlab: " + matlab);
            System.out.println("from Matlab:\n" + Matrix.parseMatlab(matlab));
            System.out.println("to Matlab:\n" + Matrix.parseMatlab(matlab).toMatlab() + "\n");
        } catch (Exception e) {
            e.printStackTrace();
        }
    }
}
